From the top of a tower, of 100m height, the angles of depression of two objects, 200 3 m apart on the horizontal plane on a line passing through the foot of the tower and on the same side of the tower are 45 ^ o - A

Correct answer: A. 15 ^ o Let OP be tower and objects are placed at A and B , then The distance between the objects = 100[ ( 45 ^ o - A) - ( 45 ^ o + A)] from OPA and OPB = 100((1 + tanA)/(1 - tanA) - (1 - tanA)/(1 + tanA)) = 100( (1 + tan A) ^ 2 - (1 - tan A) ^ 2 1 - ^ 2 A ) = 100 . 4tan A 1 - ^ 2 A = 200 2A ∴ 200 3 = 200 2A ⇒ 2A = 1 3 ∴ 2A = 30 ^ o ⇒ A = 15 ^ o

NTA Abhyas 2020

Exam: NTA Abhyas
Year: 2020
Subject: Mathematics
Chapter: Heights and Distances

NTA Abhyas 2020 · Mathematics · Heights and Distances

From the top of a tower, of 100m height, the angles of depression...

From the top of a tower, of

100m

height, the angles of depression of two objects,

\dfrac{200}{\sqrt{3}}m

apart on the horizontal plane on a line passing through the foot of the tower and on the same side of the tower are

{45}^{o} - A

and

{45}^{o} + A .

Then angle

A

is equal to From the top of a tower, of 100m height, the angles of depression of two objects, 200 3 m apart on the horizontal plane

Options

A ${15}^{o}$
B ${35}^{o}$
C $22\dfrac{{1}^{o}}{2}$
D ${45}^{o}$

Correct answer

A. ${15}^{o}$

Step-by-step solution

Let

OP

be tower and objects are placed at

A \text{ and } B

, then
The distance between the objects

= 100\left[\cot \left({45}^{o} - A\right) - \cot \left({45}^{o} + A\right)\right]

{ from\delta OPA \text{ and } \delta OPB }

= 100\left(\dfrac{1 + tanA}{1 - tanA} - \dfrac{1 - tanA}{1 + tanA}\right)

= 100\left(\dfrac{{\left(1 + tan A\right)}^{2} - {\left(1 - tan A\right)}^{2}}{1 - \tan^{2}A}\right)

= 100 . \dfrac{4tan A}{1 - \tan^{2}A} = 200\tan 2A

∴ \dfrac{200}{\sqrt{3}} = 200\tan 2A

⇒ \tan 2A = \dfrac{1}{\sqrt{3}}

∴ 2A = {30}^{o} ⇒ A = {15}^{o}

Practice Heights and Distances on Quantrex Academy →

More from Heights and Distances

The angle of elevation of the top P of a tower from the feet of one person standing due… JEE Main 2023 From the top A of a vertical wall AB of height 30m, the angles of depression of the top P… JEE Main 2023 The angle of elevation of the top of a tower from a point A due north of it is α and from… JEE Main 2022 A horizontal park is in the shape of a triangle OAB with AB = 16 . A vertical lamp post… JEE Main 2022 The angle of elevation of the top P of a vertical tower PQ of height 10 from a point A on… JEE Main 2022 Let a vertical tower AB of height 2h stands on a horizontal ground. Let from a point P on… JEE Main 2022 A tower PQ stands on a horizontal ground with base Q on the ground. The point R divides… JEE Main 2022 From the base of a pole of height 20 meter, the angle of elevation of the top of a tower… JEE Main 2022 Let AB and PQ be two vertical poles, 160m apart from each other. Let C be the middle… JEE Main 2022 A vertical pole fixed to the horizontal ground is divided in the ratio 3: 7 by a mark on… JEE Main 2021 Two poles AB of length a metres and CD of length a + b ( b ≠ a ) metres are erected at… JEE Main 2021 A spherical gas balloon of radius 16 meter subtends an angle 60 ° at the eye of the… JEE Main 2021 A pole stands vertically inside a triangular park ABC . Let the angle of elevation of the… JEE Main 2021 A man is observing, from the top of a tower, a boat speeding towards the tower from a… JEE Main 2021 The angle of elevation of a jet plane from a point A on the ground is 60 ° . After a… JEE Main 2021 Two vertical poles are 150m apart and the height of one is three times that of the other.… JEE Main 2021 Full Heights and Distances list Heights and Distances JEE Maths PYQs JEE Mathematics PYQs All NTA Abhyas JEE Main PYQs